TITLE: volume of compact simple Lie groups under the natural Euclidean embedding
QUESTION [10 upvotes]: I am looking for a quick reference for the volume formula for all the compact simple Lie groups embedded as matrix groups in the natural way. The one I care most for are the real orthogonal groups. I don't see how to deduce them from the Weyl integration formula. I believe they are of the order diameter raised to the power of the dimension. So for instance for $SO(n)$ the volume should be of order $n^{\Theta(n^2)}$.

REPLY [5 votes]: As far as I know, the most standard volume formula was obtained by I.G. Macdonald in a very short 1980 paper here.   Since the volume depends on the choice of Haar measure, Macdonald starts with the (complexified) Lie algebra of the compact Lie group, fixing a Lebesgue measure there along with a fixed lattice such as a Chevalley $\mathbb{Z}$-form.   The second ingredient in his formula comes from the standard invariants in the cohomology calculation for the group.
There is another approach to Macdonald's formula in a later paper by Y. Hashimoto:
On Macdonald’s formula for the volume of a compact Lie group., Comment. Math. Helv. 72 (1997), no. 4, 660–662.